A215197 Numbers k such that k and k + 1 are both of the form p*q^4 where p and q are distinct primes.

2511, 7856, 10287, 15471, 15632, 18063, 20816, 28592, 36368, 40816, 54512, 75248, 88047, 93231, 101168, 126927, 134703, 160624, 163376, 170991, 178767, 210032, 215216, 217808, 220624, 254096, 256527, 274671, 280624, 292976, 334448, 347408, 443151, 482192

Offset: 1

Author

Michel Lagneau, Aug 05 2012

Keywords

• nonn

Comments

The smaller of adjacent terms in A178739. - R. J. Mathar, Aug 08 2012

These are numbers n such that n and n+1 both have 10 divisors. Proof: clearly n and n+1 cannot both be of the form p^9, so for contradiction assume either n and n+1 is of the form p*q^4 and the other is of the form r^9 where p, q, and r are prime. So p*q^4 is either r^9 - 1 = (r-1)(r^2+r+1)(r^6+r^3+1) or r^9 + 1 = (r+1)(r^2-r+1)(r^6-r^3+1). But these factors are relatively prime and so cannot represent p*q^4 unless one or more factors are units. But this does not happen for r > 2, and the case r = 2 does not work since neither 511 not 513 is of the form p*q^4. - Charles R Greathouse IV, Jun 19 2016

Examples

			2511 is a member as 2511 = 31*3^4 and 2512 = 157*2^4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A005237 and A030628.

Cf. A074172, A215173.

Programs

Maple

with(numtheory):for n from 3 to 500000 do:x:=factorset(n):y:=factorset(n+1):n1:=nops(x):n2:=nops(y):if n1=2 and n2=2 then xx1:=x[1]*x[2]^4 : xx2:=x[2]*x[1]^4:yy1:=y[1]*y[2]^4: yy2:=y[2]*y[1]^4:if (xx1=n or xx2=n) and (yy1=n+1 or yy2=n+1) then printf("%a, ", n):else fi:fi:od:

Mathematica

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 4}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 4}, AppendTo[lst, n]]], {n, 3, 55000}]; lst
(* First run program for A178739 *) Select[A178739, MemberQ[A178739, # + 1] &] (* Alonso del Arte, Aug 05 2012 *)

PARI

is(n)=numdiv(n)==10 && numdiv(n+1)==10 \\ Charles R Greathouse IV, Jun 19 2016

PARI

is(n)=vecsort(factor(n)[,2])==[1,4]~ && vecsort(factor(n+1)[,2])==[1,4]~ \\ Charles R Greathouse IV, Jun 19 2016